Yes. With our Terms of Service for Educational Accounts, students of any age can work on their StoryJumper books at home. They'll login at home the same way they login from school.
If students are under 13 years old, they'll need to be in your StoryJumper class before they can use StoryJumper at home without needing a permission slip. To verify they're in your class, go to your class page and click the "People" tab.